Yes, do let us in on your findings please.

The air wouldn't all go straight down. In the case of a heli hovering in a sealed box, there would be a column of air going down under the helicopter, striking the bottom. But the air doesn't stay there, collecting. It spreads out along the bottom, then up the sides to the roof, then toward the center and down toward the low pressure area above the rotors. There is always as much mass flow of air going up as down. All of these air molecules encounter resistance as they rub against each other and the sides of the box. It's this resistance to circulation that will allow the helicopter to hover. Some degree of this type of circulation happens even when a heli is hovering outside a box. If the heli has insufficient power, the heli will not be able to keep up with the accelerating circulation and it will "settle with power". Not too likely with a high-powered electric model heli, but a real concern with full-scale helis trying to hover at high density altitude.

All of this resistance or force transfers to the box and the weight stays the same.